2016/12/21 by Vinoth Nandakumar, Nandakumar, Vinoth, Gufang Zhao +1
Mathematics · #14L35 #14M15 #22E47 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1612.06941
openalex publication_date 2016/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of \mathfraksl2, where they use singular blocks of category O for \mathfraksln and translation functors. Here we construct a positive characteristic analogue using blocks of representations of \mathfraksln over a field k of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig's conjectures for representations of Lie algebras in positive characteristic.